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On the Number of Positive Entries in the Powers of a Non-Negative Matrix

  • N. Pullman (a1)

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A real matrix A is said to be non-negative if and only if none of its entries is negative. Suppose A is an r by r non-negative matrix. We want to examine:

  • (A)The first power of A to maximize the number of positive entries in An,
  • (B)For each 1 ≤ i ≤ r the first power of A to maximize the number of positive entries in the i-th row of An.

We shall call the former first power the index of A and the latter the i-th row index of A (index (i, A)).

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References

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1. Wielandt, H., Unzerlegbare nicht negative Matrizen, Math. Zeit. 52 (1950) pp. 642648.
2. Rosenblatt, D., Asymptotic Forms of Two Classes of Matrices, Naval Logistics Quarterly 4 (1957) pp. 151165.
3. Holladay, J. C. and Varga, R.S., On Powers of Non-Negative Matrices, A. M. S. Proceedings 9 (1958) pp. 631634.
4. Dulmage, A. L. and Mendelsohn, N.S., The Exponent of a Primitive Matrix, Can. Math. Bulletin 5 (1962) pp. 241244.
5. Wedderburn, J. H. M., Boolean Linear Associative Algebra, Ann. Math. 35 (1934) pp. 185194.
6. Mann, H. B. and Ryser, H. J., Systems of Distinct Representatives, Amer. Math. Mon. 60 (1953) pp. 397401.
7. Ore, O., Graphs and Matching Theorems, Duke Math. J. 22 (1955) p. 635.
8. Ore, O., Theory of Graphs, A.M.S. Colloquium Publ. No. 38(1962).
9. Doob, J. L., Stochastic Processes, Wiley, N.Y. 1953.
10. Pták, V., On a Combinatorial Theorem and Its Application to Non-negative Matrices, Czechoslovakian Math. J., vol.8 (83), (1958) pp. 487492.
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On the Number of Positive Entries in the Powers of a Non-Negative Matrix

  • N. Pullman (a1)

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